Just about everything bad that could happen to a painting has happened to Hubert and Jan van Eyck's Adoration of the Mystic Lamb (also known as the Ghent Altarpiece). It's almost been destroyed in a fire, was nearly burned by rioting Calvinists, it's been forged, pillaged, dismembered, censored, stolen by Napoleon, hunted in the first world war, sold by a renegade cleric, then stolen repeatedly during the second world war, before being rescued by The Monuments Men, miners and a team of commando double-agents. The fact that it was the artwork the Nazis were most desperate to steal – Göring wanted it for his private collection, Hitler as the centrepiece of his citywide super-museum – has only increased its renown.

It's easy to argue that the artwork is the most influential painting ever made: it was the world's first major oil painting, and is laced with Catholic mysticism. It's almost an A to Z of Christianity – from the annunciation to the symbolic sacrifice of Christ, with the "mystic lamb" on an altar in a heavenly field, bleeding into the holy grail.

In 1934, one of its 12 panels was stolen in a heist that has never been solved, though the case is still open and new leads are followed all the time.

On 11 April of that year, Ghent police commissioner Antoine Luysterborghs pushed through a crowd at the St Bavo Cathedral that had gathered to gawk at something that was no longer there. One of the panels, depicting The Just (or Righteous) Judges, was gone. The commissioner took a quick look, and left. The missing panel – from what was already the most stolen artwork in the world – could wait. Across the street was another theft on the same night he had already been investigating: at a cheese shop.

This is just one of many bizarre twists in the story of one of the most famous art heists in history. The theft was followed quickly by a ransom demand for one million Belgian francs. As a show of good faith, the ransomer returned one of the panel's two parts (a grisaille painting of St John the Baptist). But police remained baffled.

Then a stockbroker called Arsène Goedertier had a heart attack at a Catholic political rally. He summoned his lawyer, Georges de Vos, to his deathbed. Just before he died, De Vos claimed, Goedertier whispered: "I alone know where the Mystic Lamb is. The information is in the drawer on the right of my writing table, in an envelope marked 'mutualité.'" The lawyer followed the instructions and found carbon copies of the ransom notes, plus a final, unsent note with a tantalising clue about the stolen panel's whereabouts: "[it] rests in a place where neither I, nor anybody else, can take it away without arousing the attention of the public."

But if Goedertier did steal the panel, why? The church has been defensive, and there is an air of cover-up – as well as evidence that other members of the bishopric were involved. One theory goes that a group of church members, Goedertier among them, were involved in a failed investment scheme that lost church money. Rather than admit their failure, they stole the panel and ransomed it to cover the losses. But Goedertier was wealthy and devout; it seems odd he would resort to extorting his beloved diocese.

The investigation that followed was no more thorough than Commissioner Luysterborghs's had been. De Vos failed to alert police about Goedertier's confession for a month. Eventually, after many false leads, police concluded Goedertier had been the thief. The case went cold. The panel is still missing. But to this day, a detective with the Ghent police remains assigned to it, inheriting the case from his predecessors. Progress is still being made.

The greatest strides in solving the crime have not been made by an active officer, though. Karel Mortier was chief of the Ghent police from 1974 to 1991, and fascinated with the Just Judges theft. It was a huge unsolved mystery, not just for Ghent, for Flanders, for Belgium, but for the art world. Mortier has dedicated his quiet hours to the hobby that drives him to this day: the hunt for the lost panel.

Now in his 80s, he has done more than anyone to shed light on the case. He was the first to note that Goedertier had an eye problem that meant he could barely see in the dark, much less rob a cathedral at night. He turned up information that Goedertier already had more than the million francs demanded in the ransom in his bank account. What, then, was the motivation for stealing the panel?

Mortier also suggested Goedertier could not have acted alone: the panel was taken from the altarpiece's framework, which was so high off the ground that it needed a ladder, and at least two people, to remove it. Surely, Mortier concluded, one of the four church custodians was involved, if only to provide the ladder.

Mortier's investigation met many obstacles. The church granted him access to 600 pages of archives relating to the painting – but not the period between 1934-1945. It seemed there was either a conspiracy to hide the truth, or that those involved in the investigation, the police in particular, were wildly inept. (As well as the cheese-shop fiasco, another officer began using Goedertier's typewriter, on which he'd written the ransom notes, without fingerprinting it first.)

But perhaps most impressive have been Mortier's physical discoveries. During the second world war, 10 years after the theft, Joseph Goebbels sent an art detective, Heinrich Köhn, to Ghent to find the lost Judges panel as a gift for Hitler. Köhn's investigation concluded that the panel was originally hidden on-site, but had been moved before he arrived, to keep it out of his hands. This meshes with statements from Goedertier, which Mortier learned about but the police had never noted. Apparently, Goedertier told his wife "What is misplaced is not stolen"; when his wife suspected the Germans, he replied: "I wouldn't look so far. If they'd let me search for it, I'd stay in the vicinity of the cathedral."

Mortier feels this latter statement could refer to the church community, not just the physical location of the cathedral. In 1963, he flew to Germany and tracked down Heinrich Köhn's widow. She allowed him to copy Köhn's 176-page file on his search, which showed he was convinced the panel was simply hidden inside the cathedral. On 22 May 1942, Köhn's assistant had told the bishop that the Germans were going to close the cathedral to search it.

The bishop told him to tell the then-canon, Gabriel van den Gheyn (who had also kept the painting safely hidden during the first world war), that the Germans would try and seize it – so Van Den Gheyn arranged for the whole altarpiece to be smuggled out of the cathedral by night, hidden in a junk cart, and secreted in private homes until the war ended. It's possible that Goedertier told Van Den Gheyn during confession about the theft – but that Van Den Gheyn never came forward with the information to protect it from the Nazis. Mortier reports that Max Winders, the Belgian collaborator who worked with Köhn, later said that Van Den Gheyn had found the panel. This all sounds promising, but does not explain why Van Den Gheyn, once the war was over, would not reveal the panel's whereabouts.

Mortier still receives new clues all the time. He estimates he has been contacted about 350 possible locations for the Judges panel, none of them correct. Saint Bavo has been searched six times since the second world war; Mortier even supervised an x-ray of the whole cathedral to a depth of 10 metres.

The case remains alive in the public's imagination too: in 1995, Goedertier's skull was illegally excavated by an amateur detective. Someone then stole the skull to host a seance and ask Goedertier about the theft. In 2004, Mortier ran DNA tests on the stamps of the ransom letters, reasoning that they might contain the thief's saliva. The tests were inconclusive.

The current detective in charge of the case, Jan de Kesel, made a major discovery when he tracked a lead to a church in Goedertier's home town, and found the outline of a panel the exact dimensions of the lost Judges. Was this where Van Den Gheyn hid it from the Nazis? (Though this still does not explain why Van Den Gheyn would not return the panel after the second world war.) There are gaps in every theory. The attorney general's office maintains a 2,000-page file on the theft, and the Canon of Saint Bavo still gets dozens of tip-offs that don't amount to anything – except that this theft is still aching to be solved.